"A good stack of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one." – Paul Halmos

Update

Apologies for the lack of posts recently; I’m on Spring Break and trying to relax and get some work done instead of blogging.

In other news, Stack Exchange offered me, and I accepted, an applied-mathematics internship with them this summer. (Procrastinating on math.SE finally paid off!)

I have learned some interesting things about mathematics education from seeing the types of questions which pop up most commonly on math.SE. If I can think of a way to coherently summarize my thoughts I might write a post about it.

Data mining. They’re interested in what quantitative factors predict the success of a StackExchange site, or something like that. It sounds interesting and very different from what I’ve been doing until now, so I think it’ll be a good learning experience. (And yes – my spring break is six weeks long.)

I mean that there are interesting patterns in the kinds of questions students ask most often. For example, there are a lot of questions on what matrix multiplication actually means. It seems this is just not well communicated in ordinary classes.

First, thank you for being a good teacher.
Like many, I guess, I have followed your posts for some time now and I am curious. I hope the questioning will not put you ill-at-ease.

Did you accept this applied position because you feel a lack of connection to reality/contributions to society in scholastic mathematical research? Is it more of a vacation from abstract mathematics, or do you take as a duty to work at least once in such a setting? More generally what career do you plan?

Do you feel that big problems in mathematics are already well taken care of? Do you feel that they are perhaps not all that important? Or do you have some in mind that you intend to tackle?

Also, something I have been curious about is why you did not invest more into mathematics competitions. I think you presented yourself to the Putnam in 2009. Did you also in 2010? Will you this year? I think you did not participate in olympiads. Is this the result of sort of postponing problem-solving to spend more time learning heavier/more efficient machinery? Which of olympiads or Putnam do you find more interesting? What special attract has each, if any? Any improvement/evolution to propose to them, or the mathematics competition scene in general?

Relevantly, I remember a post on automated theorem proving in polygonal geometry or something, elicited by an olympiad geometry problem.

Whatever thought you feel like sharing on the “importance of mathematics”, and other topics I mentioned here, will be scrutinized. :|

Regarding the SE position, there are a few related reasons. The first is doing something outside my comfort zone, which is something I try to do periodically in general. The second is specifically doing some mathematics outside my comfort zone, which I think will help broaden my perspective on mathematics and its relation to the world. I plan on entering academia, but I’d like to do it with a healthy awareness of the feasibility of my other options.

Regarding big problems, there are certainly a lot of them, and some of them are certainly important. I don’t think aiming for a big problem is a particularly productive way to approach mathematics at this stage (and I’ve read advice on, for example, Terence Tao’s blog that agrees with this); I would prefer to collect and organize perspectives, ideas, and tools. Regarding their importance, I think it is much more important to coherently organize what is already known in a way that makes it easier to pass on to the next generation. Unfortunately I don’t think the structure of academia particularly rewards this kind of work.

I could not participate in the Putnam in 2010 because I was (and still am) in England. I did participate in the USAMO several times. Generally I think the competition scene has its place in introducing students to interesting mathematics, but placing too much emphasis on it seems unhealthy to me. As far as my specific tastes, I don’t like time limits.

This is a doubt of mine raised by your great opinions, I think, in the 2nd paragraph. Are there many research papers aimed at “collect and organize perspectives, ideas, and tools”? Or are most of the publications of such a kind are books?

I don’t know one way or the other about the volume of such papers. All I am saying is that, as far as I can tell, the structure of academia does not directly reward this kind of output relative to others. In the first stages of an academic’s career, she has to be more concerned with producing original work, since that is what will gain her recognition, employment, and so forth. Only when she has become established in her field does she have the freedom to pursue different kinds of projects, and even then there will always be the pressure to do original work.

I guess doing the problems quickly at the competitions really depend on the amount of practice you get right? I mean, generally all the competitors were pretty slow when they began but then just become better after practice.

Practice isn’t the issue; I’ve had plenty of practice. The issue is that I have trouble quickly coming up with new angles to approach a question from if the first one doesn’t work out. Without a time limit this wouldn’t be an issue; there are ways to convince yourself that a particular approach is unlikely to work on a particular problem, but all the ones I know of are rather time-consuming.

I’m in agreement with Qiaochu here. Somehow I think these timed competitions don’t really convey a great set of values for becoming a good mathematician, for example, the need for persistence and long-range thinking which goes on for months or years.

Now that you’re coming to the end of your year in Cambridge, any chance of a little blog post on what you thought of it? I hope to sit part III the year after next, so it’d be great to hear how you found it.